A-Level Unit Test: Trigonometry Small Angle Approximations - Jethwa Maths
A-Level Unit Test: Trigonometry
Small Angle Approximations
1. Given that is small and is measured in radians, use the small angle approximations to find an approximate
value of 1-4
2 3
(3)
2.
The
diagram
shows
triangle
ABC
in
which
angle
A
=
radians,
angle
B
=
3 4
radians
and
AB
=
1
unit.
a.
Use
the
sine
rule
to
show
that
AC
=
cos
1 - sin
(3)
b. Given that is a small angle, use the result in part (i) to show that, AC 1 + p + q2, where p and q are
constants to be determined.
(3)
3.
When
is
small,
show
that
the
equation
1+sin +tan 2 2 cos 3-1
can
be
written
as
1 1-3
(4)
b.
Hence
write
down
the
value
of
1+sin +tan 2 2 cos 3-1
when
is
small.
(1)
4a. When x is small, show that tan(3x) cos(2x) can be approximated by 3x ? 6x3
(3)
b. Hence, approximate the value of tan (0.3)cos(0.2)
(2)
c. Calculate the percentage error in your approximation.
(1)
Total marks: 20
Mark Scheme
1.
4
=
1-
(4)2 2
M1
3 = 3
M1
1-4 2 3
1-[1- (42)2] (2)(3)
82 62
4 3
M1
2a.
34
=
sin
1 (-34-)
M1
AC =
34 14 cos - 14 sin
M1
3 4
=
1 4
=
1 4
AC = 1
M1
cos-sin
2b.
AC
=
(1
+
(-
-
1 2
2))-1
M1
AC
=
1
+
(-1)(-
-
?
2)
+
(-1)(-2)(-
2
-
?
2)
+
....
M1
Therefore,
AC
1
+
+
3 2
2
M1
3a.
2cos
3
2(1
-
92)
2
=
2
-
92
M1
2cos 3 - 1 1 - 92 = (1 - 3)(1 + 3)
M1
1 + sin + tan 2 = 1 + + 2 = 1 + 3
M1
1+sin +tan 2 =
1+3
=1
2 cos 3-1 (1-3)(1+3) 1-3
M1
3b.
When
is
small,
1 1-3
1
M1
4a.
tan(3x)cos(2x) = 3x(1 - (2)2)
M1
2
= 3x(1 ? 2x2)
M1
= 3x ? 6x3
M1
4b.
x = 0.1, 3(0.1) ? 6(0.1)3 = 0.294
M1
4c.
tan(0.3)cos(0.2) = 0.3031701196
M1
%
error
=
0.3031701196-0.294 0.3031701196
?
100
M1
= 3.02% (to 2 decimal places)
M1
A-Level Unit Test: Trigonometry
Proof
1. Show that the equation tan 2x = 5 sin 2x can be written in the form (1 ? 5cos 2x)sin 2x = 0
(3)
2. Prove that tan x + cot x 2cosec 2x
(4)
3.
Prove
1-cos 2 sin 2
=
tan
(3)
4.
Prove
cosec
x
+
tan
x
=
cot
2
(4)
5. Show that cosec 2x + cot 2x = cot x
(4)
6. By writing 3x = (2x + x), show that 3x = 3sin x ? 4sin3 x
(4)
7. Use the identity cos2x + sin2x = 1 to prove that tan2x = sec2x ? 1
(2)
8. Show that (sin x + tan x)(cos x + cot x) (1 + sin x)(1 + cos x)
(3)
9. Show that sec2x ? sin2x tan2x + cos2x
(2)
10.
Prove
that
1-sin 2 -2 cos
sin
x
(3)
11. Prove that sin x + sin 2x + sin 3x sin2x (2cos x + 1)
(2)
12. Prove the identity 1-cos =
1+cos
2
2
(2)
Total marks: 36
Mark Scheme
1.
tan 2x = 5sin 2x
M1
sin 2 cos 2
=
5
sin
2
sin 2x = 5sin 2x cos 2x
M1
sin 2x ? 5sin 2x cos 2x = 0
(sin 2x)(1 ? 5cos 2x) = 0 (1 ? 5 cos 2x) sin 2x = 0
M1
2.
tan x + cot x = sin + cos = 2+ 2
M1
cos sin
2+ 2 =
1
?2
M1
2
=
2
2 cos
sin
M1
=
2 2
M1
= 2 cosec 2x
3.
1-cos 2 = 1-(1-22)
M1
sin 2
2
=
1-1+22 2
= sin
M1
cos
= tan x
M1
4.
cosec x + tan x = 1 + cos
sin sin
M1
= 1+cos
sin
M1
=
1+222-1 222
=
222 222
M1
=
cos2 sin2
=
cot
2
M1
5.
cosec 2x + cot 2x = 1 + cos 2
= 1+cos 2
sin 2 sin 2
M1
sin 2
= 1+(22-1)
M1
2 cos
= 22
= 2cossin cos
M1
sin
= cot x
M1
6.
sin (3x) = sin (2x + x)
= sin 2x cos x + sin x cos 2x
M1
= 2sinx cos x (cos x) + sin x cos 2x
= 2sin x cos2x + sin x(1 ? 2sin2x)
M1
= 2 sin x(1 ? sin2x) + sin x(1 ? 2sin2 x)
M1
= 2 sin x ? 2sin3 x + sin x ? 2sin3 x = 3sin x ? 4sin3x
M1
7.
cos2x + sin2x = 1
2 2
1
M1
2 + 2 = 2
1 + tan2x = sec2x Therefore, tan2x = sec2x - 1
M1
8.
= sin x cos x + sin x cot x + tan x cos x + 1 = sin x cos x + cos x + sin x + 1
M1
= sin x(cos x + 1) + cos x + 1 = (cos x + 1)(sin x + 1)
M1
9.
(LHS) 1 + tan2x ? (1 ? cos2x)
M1
= tan2x + cos2x
M1
10.
(LHS) = sin (1-2)
sin ( -2)
M1
= sin (1-sin 2)
M1
1-2 sin cos
=
sin (1-2) 1-sin 2
M1
= sin x
11.
(LHS)
=
2
sin
+3 2
cos
-3 2
+
sin
2x
M1
= 2 sin 2x cos (-x) + sin 2x
= 2 sin 2x cos x + sin 2x
M1
= sin 2x(2 cos x + 1) (RHS)
12.
(LHS) =
=
222 222
1-(1-222) 1+(222-1)
= tan2
2
M1 M1
................
................
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